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After arriving from my one week holiday abroad, I thought I’d write a bit of a fun post inspired by an article I read on mymordernmet about math cakes.

Dinara Kasko, former architect who is now a baker, has created a series of cakes inspired by art and mathematics. Whilst Kasko employs fascinating processes to make the original cakes, they are composed of conventional ingredients. With the use of algorithmic tools and complex diagramming techniques, Kasko cakes reminisce “3D graphs, geometric models, and avant-garde sculptures.“

3D Models by Henry Segerman

Henry Segerman is an Australian mathematician who creates 3D printed models that express mathematical formulae and concepts. This allows his students to better understand them.

“The language of mathematics is often less accessible than the language of art, but I can try to translate from one to the other, producing a picture or sculpture that expresses a mathematical idea.”

Hevea Project

A French team of mathematicians called the Hevea Project, have created digital constructions of isometric embeddings.

“Take a sphere – say the surface of a tennis ball – and imagine shrinking it down to have a nanometre radius,” writes Daniel Matthews about isometric embedding. “Nash and Kuiper show that by ‘ruffling’ the surface sufficiently (but always smoothly; no creasing or folding or ripping or tearing allowed!) you can have an isometric copy of your original tennis ball, all contained within this nanometre radius.”

The De Morgan Foundation organised a one day synopsium called ‘Sublime Symmetry’ on the 13th January, which explored the mathematics behind William De Morgan’s ceramic designs.

William De Morgan was a ceramic designer in the late Victorian period.

“His conjuring of fantastical beasts to wrap themselves around the contours of ceramic hollowware and his manipulation of fanciful flora and fauna to meander across tile panels fascinated his contemporaries and still captivates today.“

The ‘Sublime Symmetry’ exhibition highlights the influence of geometry in William De Morgan’s work, and particularly the use of symmetry to create his designs. This application of geometry naturally produces beautiful and visually striking images. Below are some images of his work:

Source: auckboro.wordpress.com

I find it fascinating how mathematics is so naturally interlaced in art and beauty, and so I really wanted to share this with you. Hopefully you found it interesting as well! The ‘Sublime Symmetry’ exhibition will be at the New Walk Gallery in Leicester until the 4th March, after which it will be displayed at the William Morris Gallery in Walthamstow on the 12th March until the 3rd September. [Source]

This image is produced by visualising the foraging behaviour of the ant P. barbatus. 1000 ants stream out of the nest along 6 main trails and break off to individually search for seeds. The image only highlights 500 of their steps. The colour graduations distinguish the “search phase from the return phase and reinforce the dynamic aspects of the process”.

“Bunny” by Craig S. Kaplan and Henry Segerman

This sculpture is self-referential; it is a sculpture of a bunny whose surface is tiled by 72 copies of the word ‘bunny’. It is an example of an autologlyph, which is “a word written or represented in a way which is described by the word itself”. This type of autologlyph combines “Escher-like tessellation with typographical ideas related to ambigrams“.

Intrinsic Transformation I by Conan Chadbourne

This work shows the structure of the icosahedral group. This is the smallest non-abelian simple group and is the set of “orientation-preserving symmetries of the regular icosahedron and dodecahedron”. The groups elements are shown as yellow disks arranged at the vertices of a truncated icosahedron, whilst the group’s generators – orders 2 and 5, which are coloured red and blue respectively – are depicted by the regions between the disks.

Visualisation of the musical piece “Five Armies” by Kevin MacLeod

Musical Flocks is a project which produces animations by simulating the behaviour of agents that react to the sound of music. Slow musics results in the slow and gentle movement of the flock, whilst fast tempo music results in fast movement and abrupt changes. Sounds which have a high volumes and a “rich” frequency spectrum affect the majority of the boids, whereas low volume and less active frequencies gives “more subtle visual variations and slower graphic evolution”.

Touch-graph by Annie Verhoeven

Verhoeven used ‘thread art’ to create the sculpture shown above. It visualises a computer program that is used to show connections between people; the coloured lines symbolise the different types of connections between people. “All these different connections are used in our life on every level, be it scientific, religious, political or otherwise, and can help us determine where we stand in the world.”

Stars of the Mind’s Sky by Paul Salomon

In this image, 300 starts are positioned along concentric circles. The number of points on the star increases as the radius from the centre increases, and the stars with the same number of points are placed evenly along their circle according to the density (“jump number”) used to generate them. Mathematically, this can be seen as representing subgroups and cosets generated by elements of a cyclic group. The different colours highlight the number of cosets, with the red stars representing the generating element. Thus, “we may observe congruent stars with increasingly many cosets, shifting their way to blue along central rays through any red star.”

The Yang-Baxter scarf by Robin Endelman

The scarf above depicts the Yang-Baxter equation in statistical mechanics. A variation of this equation is used in braid theory and in the 3rd Reidemeister move in knot theory. In this scarf, the numbers 1, 2 and 3 are assigned to the colours blue, green and gold respectively. The Yang-Baxter equation states that R12 R13 R23 = R23 R13 R12, where Rij denotes the point where strand i crosses strand j;the two sides of the equation are highlighted by the two ends of the scarf, and the middle section demonstrates the equality (=).

Hyperbolic Fractal Tiling 1 by Robert Fathauer

This image shows four different views of the same 3D object – a fractal tiling where every tile has a similar dart shape. This demonstrates how a complex organic structure can be created from the repeated application of a simple set of rules to a simple starting structure.

For thousands of years, artists have used mathematical concepts in their work. In this post, I will quickly reveal some connections between these two fields throughout history.

Golden Ratio

The golden ratio is roughly equal to 1.618. The special nature of this ratio appealed to the Greeks, who thought that objects in this proportion were particularly aesthetically pleasing. It has been said that they used this ratio in their architecture and statues to ensure their beauty, for example the dimensions of the Parthenon. In fact, throughout history there have been a number of pieces of art that exhibit the golden ratio: Leonardo Da Vinci’s paintings or Michelangelo’s David. However, it has been debated whether Ancient or Renaissance artists consciously used this ratio, or whether it is simply a numerological coincidence.

Mona Lisa by Leonardo Da Vinci

Parthenon

Geometric Patterns

Geometric patterns – simple arrangements of mathematical shapes and figures – have been widely used in decoration throughout history. For example, the ‘Flower of Life’ pattern was used on the Temple of Osiris at Abydos in Egypt. Dating back about 5000 years, it consists of circles positioned in rows, each one centred on the circumference of circles in neighbouring rows.

‘Flower of Life’ Pattern

Temple of Orisis

Additionally, Mosques throughout the world are embellished with elaborate geometrical patterns, which symbolize the divine order of the Universe. The use of the geometrical patterns is due to the fact that Islamic art traditionally does not depict people and animals.

Tessellations

Popularised by Maurits Escher, tessellations are one of the more well-known and direct forms of mathematics in artwork. A tessellation is a tiling of a geometric shape with no overlaps or gaps. Escher made an art form out of colourful patterns of tessellating shapes, including reptiles, birds and fish.

Origami

Origami originated from Japan and is the craft of creating three-dimensional shapes solely by folding paper (usually only one sheet). These shapes range from paper cranes to flowers. If you unfold the piece of paper, there will be a complex geometrical pattern of creases that are made up of triangles and squares. Many of these will be congruent due to the fact that the same fold produced them, revealing the deep links between geometry and ancient art.

Origami Paper Crane

Unfolded Origami

Fractals

Fractals are mathematical structures that have the property of ‘self-similarity’, meaning that if you zoom in on one, the same type of structure will keep appearing. I have already talked about extensively in a previous blog post; check it out if you’re interested! (Personally, I find them beautiful).

Mathematics as Art

The mathematician Jerry King stated, “the keys to mathematics are beauty and elegance and not dullness and technicality”. In ‘A Mathematician’s Apology’ by G.H. Hardy, Hardy explores this idea by explaining his thoughts on the criteria for mathematical beauty: “seriousness, depth, generality, unexpectedness, inevitability, and economy”. Furthermore, Paul Erdos agreed that mathematics had beauty by explaining: “”Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful.”

If this topic interests you, I would highly recommend reading this article in AMS’s Feature Column. M x